Algebraic Preliminaries

Paul C. Eklof    Department of Mathematics, University of California, Irvine CA, U.S.A.

Alan H. Mekler    Department of Mathematics and Statistics Simon Fraser University

In the first two sections of this chapter we review the algebraic background which is assumed in the rest of the book; this also gives us the opportunity to fix notation and conventions. In the last section, we discuss linear topologies on modules. We assume the reader is already familiar with most of the material in this chapter, so it is presented informally and largely without proofs; for more on the topics covered we refer the reader to such texts as Fuchs 1970/1973, Anderson-Fuller 1992, Rotman 1979, or Weibel 1994, as well as any standard graduate text in algebra.

All rings in this book will have a multiplicative identity and all modules will be unitary modules. Unless otherwise specified, “module” will mean left R-module. Much of the time we will focus on abelian groups, that is -modules, and we will often refer to these simply as groups.

If φ: A → B is a function, and X ⊆ A, φ[X] = {φ(a): a ∈ X};φ[A] will also be denoted im(φ) or rge(φ). If Y ⊆ B, φ- 1[Y], = {a ∈ A: φ(a) ∈ Y}; if φ is a homomorphism, ker(φ) = φ_1[{0}]. The restriction of φ to X is denoted φ X, i.e., φ X = {(x, φ(x)): x ∈ X}. In an abuse of notation, sometimes we will write M = 0 instead of M = {0} and φ–1[x] instead of φ–1[{x}]. If ψ: B → C, ψ φ is the composition of ψ with φ, a function from A to C.

If M is a module and Y ⊆ M, then (Y) denotes the submodule generated by Y. The notation Y ⊂ X means Y ⊆ X and Y ≠ 1.

1 Homomorphisms and extensions

If M and H are left R-modules, Homr(M,H) denotes the group of R-homomorphisms from M to H, which is an abelian group under the operation defined by: (f + g)(x) = f(x) + g(x). We will sometimes refer to HomR(M, H) as the H-dual of M. Often we will write Hom (M, H), if R is clear from context. If x ∈ M and y ∈ Hom(M,H), we denote by 〈y, x〉 or 〈x, y〉, interchangeably, the element y(x) of H, i.e., the result of applying y to x.

If H is an R-S bimodule (that is, a left R-module and a right S-module such that (ra)s = r(as) for all r ∈ R, a ∈ H, s ∈ S), then Homr(M,H) has a right S-module structure defined by: (fs)(x) = f(x)s. The bimodule structure that we will be interested in arises as follows. If H is an R-module, let Endr(H) = Homr(H, H); then EndR(H) is a ring under composition of homomorphisms, where we define the product f ⋅ g to be g f; H has a right EndR(H)-module structure defined by: af = f(a) for all a ∈ H and f ∈ Endr(H). This makes H into an R-EndR(H)-bimodule. Note that EndR(5) ≅ R.

If M is an abelian group, then M*, without further explanation, will denote ℤ(M,?ℤ), called the dual group of M. A group is called a dual group if and only if it is of the form Hom(M, ) for some group M. The structure and properties of dual groups will be one of the principal subjects of this book. In particular, we will be interested in when a group is (canonically) isomorphic to its double dual. Sometimes it will be convenient to consider this question in the more general context of H-duals.

Let us temporarily fix an R-module H, and let S denote EndR(H), so that H is an R-S-bimodule. For convenience denote Homr(M,H) by M*. Then M* is a right S-module, and HomS(M*, H) is a left R-module in the obvious fashion; we denote the latter by M**. There is a canonical homomorphism

M:M→M**

defined by: 〈σM(x),y〉 = 〈x,y〉 for all x ∈ M and y ∈ M*. We say that M is H-torsionless if σM is one-one, and that M is H-reflexive if σM is one-one and onto M, i.e., an isomorphism. If H = R, we say torsionless or reflexive instead of R-torsionless or R-reflexive, respectively.

For every R-homomorphism φ: M → N there is an induced S-homomorphism φ* : Hom(N,H) → Hom(M,H) defined by: φ*(f) = f φ for all f ∈ Hom(N,H). There is also an induced S-homomorphism φ*: Hom(H, M) → Hom(H, N) defined by: φ*(g) = φ gfor all g ∈ Hom(H, M). If φ is an isomorphism, then so are φ* and φ*. If φ is surjective, then φ* is injective; if φ is injective, then φ* is injective. But φ* may not be surjective when φ is injective; and φ* may not be surjective when φ is surjective. In fact, we have the following situation. A sequence of homomorphisms

→Mn−1→φn−1Mn→φnMn+1→…

is called exact if ker(φn) = im(φn-1) for all n. A short exact sequence (or s.e.s.) is an exact sequence of the form

→L→ψM→φN→0.

Given a short exact sequence of R-homomorphisms as above and given an R-module H, the sequences

→Hom(N,H)→φ*Hom(M,H)→ψ*Hom(L,H)0→Hom(H,L)→ψ*Hom(H,M)→φ*Hom(H,N)

are exact. We have the following fundamental theorem of Cartan-Eilenberg 1956. (In its statement we will ignore the complication that the domain of the function is a proper class.)